We carried our snacks in little plastic containers and ziploc bags in a backpack. We had carrots, apples, nuts/seeds, fruit leather/dried fruit, pretzels, and pbj sandwiches (I cut them with a dinosaur shaped cutter and used whole wheat bread, so they actually didn't get too smooshed/soggy).
Also, we used hydration backpacks. Your kids might prefer those (not sure if you were going to carry water bottles or buy drinks in the parks), plus the ice water will keep your snacks cool throughout the day.